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Mirrors > Home > MPE Home > Th. List > canthwdom | Structured version Visualization version GIF version |
Description: Cantor's Theorem, stated using weak dominance (this is actually a stronger statement than canth2 8654, equivalent to canth 7090). (Contributed by Mario Carneiro, 15-May-2015.) |
Ref | Expression |
---|---|
canthwdom | ⊢ ¬ 𝒫 𝐴 ≼* 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0elpw 5221 | . . . . 5 ⊢ ∅ ∈ 𝒫 𝐴 | |
2 | ne0i 4250 | . . . . 5 ⊢ (∅ ∈ 𝒫 𝐴 → 𝒫 𝐴 ≠ ∅) | |
3 | 1, 2 | mp1i 13 | . . . 4 ⊢ (𝒫 𝐴 ≼* 𝐴 → 𝒫 𝐴 ≠ ∅) |
4 | brwdomn0 9017 | . . . 4 ⊢ (𝒫 𝐴 ≠ ∅ → (𝒫 𝐴 ≼* 𝐴 ↔ ∃𝑓 𝑓:𝐴–onto→𝒫 𝐴)) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (𝒫 𝐴 ≼* 𝐴 → (𝒫 𝐴 ≼* 𝐴 ↔ ∃𝑓 𝑓:𝐴–onto→𝒫 𝐴)) |
6 | 5 | ibi 270 | . 2 ⊢ (𝒫 𝐴 ≼* 𝐴 → ∃𝑓 𝑓:𝐴–onto→𝒫 𝐴) |
7 | relwdom 9014 | . . . . 5 ⊢ Rel ≼* | |
8 | 7 | brrelex2i 5573 | . . . 4 ⊢ (𝒫 𝐴 ≼* 𝐴 → 𝐴 ∈ V) |
9 | foeq2 6562 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑓:𝑥–onto→𝒫 𝑥 ↔ 𝑓:𝐴–onto→𝒫 𝑥)) | |
10 | pweq 4513 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴) | |
11 | foeq3 6563 | . . . . . . . 8 ⊢ (𝒫 𝑥 = 𝒫 𝐴 → (𝑓:𝐴–onto→𝒫 𝑥 ↔ 𝑓:𝐴–onto→𝒫 𝐴)) | |
12 | 10, 11 | syl 17 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑓:𝐴–onto→𝒫 𝑥 ↔ 𝑓:𝐴–onto→𝒫 𝐴)) |
13 | 9, 12 | bitrd 282 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑓:𝑥–onto→𝒫 𝑥 ↔ 𝑓:𝐴–onto→𝒫 𝐴)) |
14 | 13 | notbid 321 | . . . . 5 ⊢ (𝑥 = 𝐴 → (¬ 𝑓:𝑥–onto→𝒫 𝑥 ↔ ¬ 𝑓:𝐴–onto→𝒫 𝐴)) |
15 | vex 3444 | . . . . . 6 ⊢ 𝑥 ∈ V | |
16 | 15 | canth 7090 | . . . . 5 ⊢ ¬ 𝑓:𝑥–onto→𝒫 𝑥 |
17 | 14, 16 | vtoclg 3515 | . . . 4 ⊢ (𝐴 ∈ V → ¬ 𝑓:𝐴–onto→𝒫 𝐴) |
18 | 8, 17 | syl 17 | . . 3 ⊢ (𝒫 𝐴 ≼* 𝐴 → ¬ 𝑓:𝐴–onto→𝒫 𝐴) |
19 | 18 | nexdv 1937 | . 2 ⊢ (𝒫 𝐴 ≼* 𝐴 → ¬ ∃𝑓 𝑓:𝐴–onto→𝒫 𝐴) |
20 | 6, 19 | pm2.65i 197 | 1 ⊢ ¬ 𝒫 𝐴 ≼* 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 = wceq 1538 ∃wex 1781 ∈ wcel 2111 ≠ wne 2987 Vcvv 3441 ∅c0 4243 𝒫 cpw 4497 class class class wbr 5030 –onto→wfo 6322 ≼* cwdom 9012 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fo 6330 df-fv 6332 df-wdom 9013 |
This theorem is referenced by: pwdjudom 9627 |
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